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Due to the characteristic of risk aversion, option has become one of the most fashionable derivatives in the financial field. More and more investigators are attracted to devote themselves to exploring the option pricing problem. In this paper, we are concerned with the valuation of American lookback options in terms of the Black-Scholes model. It is well known that the American lookback option satisfies a two-dimensional nonlinear partial differential equation in an unbounded domain, which couldn't be numerically solved directly. Based on the analysis of the issues for solving this problem, this paper introduces an approach to settle it. First, we transform the problem into a one-dimensional form by the numeraire transformation. And then, the Landau's transformation is applied to normalize the defined domain. For the nonlinear feature of the resulting problem, we propose a finite volume method coupled with Newton iterative method to obtain the optional value and the optimal exercise boundary simultaneously. We also give a proof on the nonnegativity of the numerical solutions under some appropriate assumptions. Finally, some numerical simulations are presented using the proposed method in this paper. Comparing with the binomial method, we can conclude that the proposed method is an effective one, which provides a theoretical basis for practical applications.
[1] Vogel E E, Saravia G 2014 Eur. Phys. J. B 87 177
[2] Ibuki T, Suzuki S, Inoue J 2013 Econophysics of systemic risk and network dynamics (Milan:Springer) pp239-259
[3] Mimkes J 2012 Continuum Mech. Thermodyn. 24 731
[4] Chakraborti A, Muni T I, Patriarca M, Abergel F 2011 Fuant. Financ. 11 1013
[5] Sousa T, Domingos T 2006 Phys. A 371 492
[6] Fang H 2014 Acta Phys. Sin. 63 038902 (in Chinese) [范宏 2014 物理学报 63 038902]
[7] Da C, Fan H Y 2014 Acta Phys. Sin. 63 098901 (in Chinese) [笪诚, 范洪义 2014 物理学报 63 098901]
[8] Yu Z R 2000 Physics 29 662 (in Chinese) [于祖荣 2000 物理 29 662]
[9] Kwok Y K 2008 Mathematical models of financial derivatives (Vol. 2) (Berlin:Springer) pp201-211
[10] Zhang R, Song H M, Luan N N 2014 Front. Math. China 9 455
[11] Kim K Ik, Park H S, Qian X S 2011 J. Comput. Appl. Math. 235 5140
[12] Zhang T, Zhang S H, Zhu D M 2009 J. Comput. Math. 27 484
[13] Li G, Zhu B X, Zhang Q, Song H M 2014 Journal of Jilin University(Science Edition) 52 698 (in Chinese) [李庚, 朱本喜, 张琪, 宋海明 2014 吉林大学学报(理学版) 52 698]
[14] Wang H, Basu T S 2012 SIAM J. Sci. Comput. 34 A2444
[15] Han Q G, Ma H A, Xiao H Y, Li R, Zhang C, Li Z C, Tian Y, Jia X P 2010 Acta Phys. Sin. 59 1923 (in Chinese) [韩奇钢, 马红安, 肖宏宇, 李瑞, 张聪, 李战厂, 田宇, 贾晓鹏 2010 物理学报 59 1923]
[16] Kwon Y H, Lee Y 2011 SIAM J. Numer. Anal. 49 2598
[17] Mattsson K, Carpenter M H 2010 SIAM J. Sci. Comput. 32 2298
[18] Forsyth P A, Vetzal K R, Zvan R 1999 Appl. Math. Finance 6 87
[19] Li H Y, Ma H P, Sun W W 2013 SIAM J. Numer. Anal. 51 353
[20] Pindza E, Patidar K C, Ngounda E 2014 Numer. Methods Partial Differential Equations 30 1169
[21] Chen Y P, Huang F L, Yi N Y, Liu W B 2011 SIAM J. Numer. Anal. 49 1625
[22] Bessemoulin C M, Filbet F 2012 SIAM J. Sci. Comput. 34 B559
[23] Zhang K, Wang S 2008 Appl. Math. Comput. 201 398
[24] Berton J, Eymard R 2006 MSAIN Math. Model. Numer. Anal. 40 311
[25] Angermann L, Wang S 2007 Numer. Math. 106 1
[26] Jiang L S 2007 Mathematical modeling and methods of option pricing (Vol. 2) (Beijing:Higher Education Press) p303 (in Chinese) [姜礼尚 2007 期权定价的数学模拟和方法(第二版)(北京:高等教育出版社)第303页]
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[1] Vogel E E, Saravia G 2014 Eur. Phys. J. B 87 177
[2] Ibuki T, Suzuki S, Inoue J 2013 Econophysics of systemic risk and network dynamics (Milan:Springer) pp239-259
[3] Mimkes J 2012 Continuum Mech. Thermodyn. 24 731
[4] Chakraborti A, Muni T I, Patriarca M, Abergel F 2011 Fuant. Financ. 11 1013
[5] Sousa T, Domingos T 2006 Phys. A 371 492
[6] Fang H 2014 Acta Phys. Sin. 63 038902 (in Chinese) [范宏 2014 物理学报 63 038902]
[7] Da C, Fan H Y 2014 Acta Phys. Sin. 63 098901 (in Chinese) [笪诚, 范洪义 2014 物理学报 63 098901]
[8] Yu Z R 2000 Physics 29 662 (in Chinese) [于祖荣 2000 物理 29 662]
[9] Kwok Y K 2008 Mathematical models of financial derivatives (Vol. 2) (Berlin:Springer) pp201-211
[10] Zhang R, Song H M, Luan N N 2014 Front. Math. China 9 455
[11] Kim K Ik, Park H S, Qian X S 2011 J. Comput. Appl. Math. 235 5140
[12] Zhang T, Zhang S H, Zhu D M 2009 J. Comput. Math. 27 484
[13] Li G, Zhu B X, Zhang Q, Song H M 2014 Journal of Jilin University(Science Edition) 52 698 (in Chinese) [李庚, 朱本喜, 张琪, 宋海明 2014 吉林大学学报(理学版) 52 698]
[14] Wang H, Basu T S 2012 SIAM J. Sci. Comput. 34 A2444
[15] Han Q G, Ma H A, Xiao H Y, Li R, Zhang C, Li Z C, Tian Y, Jia X P 2010 Acta Phys. Sin. 59 1923 (in Chinese) [韩奇钢, 马红安, 肖宏宇, 李瑞, 张聪, 李战厂, 田宇, 贾晓鹏 2010 物理学报 59 1923]
[16] Kwon Y H, Lee Y 2011 SIAM J. Numer. Anal. 49 2598
[17] Mattsson K, Carpenter M H 2010 SIAM J. Sci. Comput. 32 2298
[18] Forsyth P A, Vetzal K R, Zvan R 1999 Appl. Math. Finance 6 87
[19] Li H Y, Ma H P, Sun W W 2013 SIAM J. Numer. Anal. 51 353
[20] Pindza E, Patidar K C, Ngounda E 2014 Numer. Methods Partial Differential Equations 30 1169
[21] Chen Y P, Huang F L, Yi N Y, Liu W B 2011 SIAM J. Numer. Anal. 49 1625
[22] Bessemoulin C M, Filbet F 2012 SIAM J. Sci. Comput. 34 B559
[23] Zhang K, Wang S 2008 Appl. Math. Comput. 201 398
[24] Berton J, Eymard R 2006 MSAIN Math. Model. Numer. Anal. 40 311
[25] Angermann L, Wang S 2007 Numer. Math. 106 1
[26] Jiang L S 2007 Mathematical modeling and methods of option pricing (Vol. 2) (Beijing:Higher Education Press) p303 (in Chinese) [姜礼尚 2007 期权定价的数学模拟和方法(第二版)(北京:高等教育出版社)第303页]
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